Critical configurations for three projective views

TL;DR

This work provides an algebraic framework to classify critical configurations in structure-from-motion with three projective cameras. By blowing up the 3-space at camera centers and examining the intersections of multi-view varieties, the authors reduce maximal critical configurations to the study of compatible triples of quadrics and their ambiguous points, yielding a complete taxonomy of surface-, curve-, and finite-point-based configurations. A central result is that, for eight or more points, criticality occurs if and only if the points lie on the strict transform of one of the described varieties (or on a subset), with six or fewer points always critical; a notable new configuration is twist ed cubic with a secant line. The approach generalizes two-view results through a careful analysis of quadric intersections, ultimately producing a detailed catalog of all maximal three-view critical configurations and their conjugates, with implications for the stability and identifiability of three-view reconstructions in practice.

Abstract

The problem of structure from motion is concerned with recovering the 3-dimensional structure of an object from a set of 2-dimensional images taken by unknown cameras. Generally, all information can be uniquely recovered if enough images and point correspondences are provided, yet there are certain cases where unique recovery is impossible; these are called critical configurations. We use an algebraic approach to study the critical configurations for three projective cameras. We show that all critical configurations lie on the intersection of quadric surfaces, and classify exactly which intersections constitute a critical configuration.

Figures (8)

• Figure 1: The blow-down of some of the critical configurations for three views. [1] The rational quartic may have a node or cusp. [2] The twisted cubic may degenerate to the union of a conic and a line, or to three lines. [3] The 7-point configuration must satisfy the conditions in \ref{['prop:7_points_critical']}.
• Figure 2: Illustration showing all critical quadrics with two marked camera centers, i.e. those containing a pair of permissible lines. The ones marked in red are not critical and do not contain such a pair of lines.
• Figure 3: If the camera centers are positioned as shown, the three quadrics (all three equal) do not constitute a compatible triple.
• Figure 4: One example of six lines satisfying the conditions in \ref{['thr:compatible_quadrics']}
• Figure 5: Given three points $p_1',p_2',p_3'$ on a smooth cubic plane curve, there exist three lines forming a triangle with vertices $y_1,y_2,y_3$ such that each line contains exactly one point.

Theorems & Definitions (108)

• Theorem 1.1
• Remark 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Theorem 2.7: Tomas
• Definition 2.8
• Definition 2.9